Logic without content? Takeuti and Zaring

Question

I am reading the book "Introduction to Axiomatic Set Theory" by Takeuti and Zaring, and I wonder if I understand the "language of logic" from chapter 2 properly.

They write:

The language of our theory consists of
- individual variables: $x_0, x_1,\ldots,$
- a predicate constant $\in$.
- logical symbols: $\lnot, \lor, \land, \rightarrow, \leftrightarrow, \forall, \exists,$
- auxiliary symbols: $(\ ), [\ ]$.

What is confusing me is that they do not say what the meaning of any part of this language is. For instance, what kind of objects are the individual variables? How am I to understand the notion "$x_0\in x_1$"? And so on. Ultimately the individual variables are probably going to be sets, but there is no mention of sets at this point.

They later introduce the notion of "wffs" (well-formed formulas) as follows:
- If $x$ and $y$ are individual variables then $x\in y$ is a wff.
- If $\phi$ and $\psi$ are wffs then $\lnot\phi, \phi\lor\psi, \phi\land\psi, \phi\rightarrow\psi$ and $\phi\leftrightarrow\psi$ are wffs.
- If $\phi$ is a wff and $x$ is an individual variable then $(\forall x)\phi$ and $(\exists x)\phi$ are wffs.

I try to answer my own confusion as follows, and I'd appreciate any further input: The language can stand as it is without having any meaning. It just contains some collection $x_0, x_1,\ldots$ of "things" (not necessarily countable) that we dub "individual variables", and it is unimportant where they come from. The important part is how we can combine the individual variables with the given symbols to obtain "legal expressions", and this is again independent of whichever meaning we can think of for the given symbols.

In short, the rules of the language are independent of any interpretation of its components.

Is this a correct explanation?

I can form some expression like $(x_1\in x_2)\land (x_3 \in x_4)$ without having the slightest idea what it means?

And maybe even worse, without having any idea how I can check if it's true or false?

Does it even make sense to ask if it is true or false?

I apologise for the somewhat "fuzzy" question, but I've tried to make clear the points of confusion, and I have a feeling that it can all be answered objectively.

Answer 1

What we need is the concept of interpretation: the interpretation is the way to "convey" the content into a formal system.

The first step is to pick up a domain (or universe), i.e. an "environment" of object.

The variables of the language will be used to refer to objects of the domain.

If we formalize e.g. elementary geometry, the variable will be used to refer to points.

In set theory, the variables will refer to sets.

Thus, a simple formula like:

$\exists x_1 \ \lnot \exists x_2 \ (x_2 \in x_1)$

must be read as:

"there is a set ($x_1$) such that no set belongs to it"

i.e. "there is an empty set."

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Of course, when we formalize a theory, like arithmetic or set theory, we have in mind a "specific" interpretation, usually called "the intended one", but we can interpret a formal system in many ways.

The axioms of the theory must be true in that interpretation and consequently all theorems are true also in it (the theorems, being derived by the rules of logic from the axioms, are logical consequences of the axiom; thus, they are true in every interpretation satisfying the axiom).

But we can try with "alternative" interpretations.

If we consider e.g. as domain the set $\mathbb N$ of natural number and we interpret the binary predicate symbols $\in$ with the relation "less than", we can easily verify that the above formula is true also in this new interpretation, because now it "means":

$\exists x_1 \ \lnot \exists x_2 \ (x_2 < x_1)$

and the number zero satisfy it.

Of course, this will not be true for other axioms of set theory ...